Very carefully!
And by wicking CA glue along the length of the wire. I take a large sewing needle and use a grinding bit in my Dremel to grind off the top part of the "eye," which leaves an open-ended "U" shape. Then I apply a small drop of CA to the end of the needle... the glue kind of sits there filling the "U"... then I just touch that end of the needle to the wire (on the wood side of the wire, not the upholstery side) and the CA is drawn into the joint. Repeat that maybe 2-3 times along the entire length of the wire to make sure that there's CA along the whole joint.
If any CA was showing, I used very fine sandpaper to sand it off the wood and the wire, and finally I brushed some Future onto the wood to give it a little gloss.
BTW... that needle trick is what I use all the time when I want to have extreme control and precision as to where the CA needs to go.